The aim of this manuscript is to study the Leslie–Gower predator-prey model with weak Allee effect on prey, that is (REF ) with \(m<0\) . We complement the results of Ostfeld and Canhan [1]} where the authors studied the stabilisation of file-vole population which depends on the variation in reproductive rate and recruitment of the population (Allee effect). File-vole specie also depends on the survival rate for adults were delayed density-dependent (weak Allee effect). On the other hand, we observe that the main difference between system (REF ) and (REF ) with \(m<0\)  is the fact that (REF ) has at most three positive equilibrium points in the first quadrant instead of one for system (REF ) and two for system (REF ) and \(m\le 0\)  [2]}, [3]}, [4]}. This additional equilibrium points gives rise to different type of bifurcations such as saddle-node bifurcation, Bogadonov–Takens bifurcation, and homoclinic bifurcation. This manuscript also extends some of the results obtained by Arancibia–Ibarra and González–Olivares [3]} and González–Olivares et al.  [4]} for a modified Leslie–Gower model with \(m=0\) , that is, with a specific type of weak Allee effect. The authors showed the existence of a stable limit cycle which represent the oscillation of both populations.
